Deep Probabilistic Modelling with with Gaussian Processes

Abstract

Neural network models are algorithmically simple, but mathematically complex. Gaussian process models are mathematically simple, but algorithmically complex. In this tutorial we will explore Deep Gaussian Process models. They bring advantages in their mathematical simplicity but are challenging in their algorithmic complexity. We will give an overview of Gaussian processes and highlight the algorithmic approximations that allow us to stack Gaussian process models: they are based on variational methods. In the last part of the tutorial will explore a use case exemplar: uncertainty quantification. We end with open questions.

What is Machine Learning?

where data is our observations. They can be actively or passively acquired (meta-data). The model contains our assumptions, based on previous experience. That experience can be other data, it can come from transfer learning, or it can merely be our beliefs about the regularities of the universe. In humans our models include our inductive biases. The prediction is an action to be taken or a categorization or a quality score. The reason that machine learning has become a mainstay of artificial intelligence is the importance of predictions in artificial intelligence. The data and the model are combined through computation.

In practice we normally perform machine learning using two functions. To combine data with a model we typically make use of:

a prediction function a function which is used to make the predictions. It includes our beliefs about the regularities of the universe, our assumptions about how the world works, e.g. smoothness, spatial similarities, temporal similarities.

an objective function a function which defines the cost of misprediction. Typically it includes knowledge about the world's generating processes (probabilistic objectives) or the costs we pay for mispredictions (empiricial risk minimization).

The combination of data and model through the prediction function and the objectie function leads to a learning algorithm. The class of prediction functions and objective functions we can make use of is restricted by the algorithms they lead to. If the prediction function or the objective function are too complex, then it can be difficult to find an appropriate learning algorithm. Much of the acdemic field of machine learning is the quest for new learning algorithms that allow us to bring different types of models and data together.

Artificial Intelligence

Uncertainty

In practice, we normally also have uncertainty associated with these functions. Uncertainty in the prediction function arises from

scarcity of training data and

mismatch between the set of prediction functions we choose and all possible prediction functions.

There are also challenges around specification of the objective function, but for we will save those for another day. For the moment, let us focus on the prediction function.

Neural Networks and Prediction Functions

Neural networks are adaptive non-linear function models. Originally, they were studied (by McCulloch and Pitts (McCulloch and Pitts, 1943)) as simple models for neurons, but over the last decade they have become popular because they are a flexible approach to modelling complex data. A particular characteristic of neural network models is that they can be composed to form highly complex functions which encode many of our expectations of the real world. They allow us to encode our assumptions about how the world works.

We will return to composition later, but for the moment, let's focus on a one hidden layer neural network. We are interested in the prediction function, so we'll ignore the objective function (which is often called an error function) for the moment, and just describe the mathematical object of interest

Where in this case \(\mappingFunction(\cdot)\) is a scalar function with vector inputs, and \(\activationVector(\cdot)\) is a vector function with vector inputs. The dimensionality of the vector function is known as the number of hidden units, or the number of neurons. The elements of this vector function are known as the activation function of the neural network and \(\mappingMatrixTwo\) are the parameters of the activation functions.

Relations with Classical Statistics

In statistics activation functions are traditionally known as basis functions. And we would think of this as a linear model. It's doesn't make linear predictions, but it's linear because in statistics estimation focuses on the parameters, \(\mappingMatrix\), not the parameters, \(\mappingMatrixTwo\). The linear model terminology refers to the fact that the model is linear in the parameters, but it is not linear in the data unless the activation functions are chosen to be linear.

Adaptive Basis Functions

The first difference in the (early) neural network literature to the classical statistical literature is the decision to optimize these parameters, \(\mappingMatrixTwo\), as well as the parameters, \(\mappingMatrix\) (which would normally be denoted in statistics by \(\boldsymbol{\beta}\))1.

In this tutorial, we're going to go revisit that decision, and follow the path of Radford Neal (Neal, 1994) who, inspired by work of David MacKay (MacKay, 1992) and others did his PhD thesis on Bayesian Neural Networks. If we take a Bayesian approach to parameter inference (note I am using inference here in the classical sense, not in the sense of prediction of test data, which seems to be a newer usage), then we don't wish to fit parameters at all, rather we wish to integrate them away and understand the family of functions that the model describes.

Probabilistic Modelling

This Bayesian approach is designed to deal with uncertainty arising from fitting our prediction function to the data we have, a reduced data set.

The Bayesian approach can be derived from a broader understanding of what our objective is. If we accept that we can jointly represent all things that happen in the world with a probability distribution, then we can interogate that probability to make predictions. So, if we are interested in predictions, \(\dataScalar_*\) at future points input locations of interest, \(\inputVector_*\) given previously training data, \(\dataVector\) and corresponding inputs, \(\inputMatrix\), then we are really interogating the following probability density, \[
p(\dataScalar_*|\dataVector, \inputMatrix, \inputVector_*),
\] there is nothing controversial here, as long as you accept that you have a good joint model of the world around you that relates test data to training data, \(p(\dataScalar_*, \dataVector, \inputMatrix, \inputVector_*)\) then this conditional distribution can be recovered through standard rules of probability (\(\text{data} + \text{model} \rightarrow \text{prediction}\)).

We can construct this joint density through the use of the following decomposition: \[
p(\dataScalar_*|\dataVector, \inputMatrix, \inputVector_*) = \int p(\dataScalar_*|\inputVector_*, \mappingMatrix) p(\mappingMatrix | \dataVector, \inputMatrix) \text{d} \mappingMatrix
\]

where, for convenience, we are assuming all the parameters of the model are now represented by \(\parameterVector\) (which contains \(\mappingMatrix\) and \(\mappingMatrixTwo\)) and \(p(\parameterVector | \dataVector, \inputMatrix)\) is recognised as the posterior density of the parameters given data and \(p(\dataScalar_*|\inputVector_*, \parameterVector)\) is the likelihood of an individual test data point given the parameters.

The likelihood of the data is normally assumed to be independent across the parameters, \[
p(\dataVector|\inputMatrix, \mappingMatrix) \prod_{i=1}^\numData p(\dataScalar_i|\inputVector_i, \mappingMatrix),\]

The likelihood is also where the prediction function is incorporated. For example in the regression case, we consider an objective based around the Gaussian density, \[
p(\dataScalar_i | \mappingFunction(\inputVector_i)) = \frac{1}{\sqrt{2\pi \dataStd^2}} \exp\left(-\frac{\left(\dataScalar_i - \mappingFunction(\inputVector_i)\right)^2}{2\dataStd^2}\right)
\]

In short, that is the classical approach to probabilistic inference, and all approaches to Bayesian neural networks fall within this path. For a deep probabilistic model, we can simply take this one stage further and place a probability distribution over the input locations, \[
p(\dataVector_*|\dataVector) = \int p(\dataVector_*|\inputMatrix_*, \parameterVector) p(\parameterVector | \dataVector, \inputMatrix) p(\inputMatrix) p(\inputMatrix_*) \text{d} \parameterVector \text{d} \inputMatrix \text{d}\inputMatrix_*
\] and we have unsupervised learning (from where we can get deep generative models).

Graphical Models

One way of representing a joint distribution is to consider conditional dependencies between data. Conditional dependencies allow us to factorize the distribution. For example, a Markov chain is a factorization of a distribution into components that represent the conditional relationships between points that are neighboring, often in time or space. It can be decomposed in the following form. \[p(\dataVector) = p(\dataScalar_\numData | \dataScalar_{\numData-1}) p(\dataScalar_{\numData-1}|\dataScalar_{\numData-2}) \dots p(\dataScalar_{2} | \dataScalar_{1})\]

By specifying conditional independencies we can reduce the parameterization required for our data, instead of directly specifying the parameters of the joint distribution, we can specify each set of parameters of the conditonal independently. This can also give an advantage in terms of interpretability. Understanding a conditional independence structure gives a structured understanding of data. If developed correctly, according to causal methodology, it can even inform how we should intervene in the system to drive a desired result (Pearl, 1995).

However, a challenge arise when the data becomes more complex. Consider the graphical model shown below, used to predict the perioperative risk of C Difficile infection following colon surgery (Steele et al., 2012).

To capture the complexity in the interelationship between the data the graph becomes more complex, and less interpretable.

Performing Inference

As far as combining our data and our model to form our prediction, the devil is in the detail. While everything is easy to write in terms of probability densities, as we move from \(\text{data}\) and \(\text{model}\) to \(\text{prediction}\) there is that simple \(\xrightarrow{\text{compute}}\) sign, which is now burying a wealth of difficulties. Each integral sign above is a high dimensional integral which will typically need approximation. Approximations also come with computational demands. As we consider more complex classes of functions, the challenges around the integrals become harder and prediction of future test data given our model and the data becomes so involved as to be impractical or impossible.

Statisticians realized these challenges early on, indeed, so early that they were actually physicists, both Laplace and Gauss worked on models such as this, in Gauss's case he made his career on prediction of the location of the lost planet (later reclassified as a asteroid, then dwarf planet), Ceres. Gauss and Laplace made use of maximum a posteriori estimates for simplifying their computations and Laplace developed Laplace's method (and invented the Gaussian density) to expand around that mode. But classical statistics needs better guarantees around model performance and interpretation, and as a result has focussed more on the linear model implied by \[
\mappingFunction(\inputVector) = \left.\mappingVector^{(2)}\right.^\top \activationVector(\mappingMatrix_1, \inputVector)
\]

The Gaussian likelihood given above implies that the data observation is related to the function by noise corruption so we have, \[
\dataScalar_i = \mappingFunction(\inputVector_i) + \noiseScalar_i,
\] where \[
\noiseScalar_i \sim \gaussianSamp{0}{\dataStd^2}
\] and while normally integrating over high dimensional parameter vectors is highly complex, here it is trivial. That is because of a property of the multivariate Gaussian.

Multivariate Gaussian Properties

Gaussian processes are initially of interest because

linear Gaussian models are easier to deal with

Even the parameters within the process can be handled, by considering a particular limit.

Let's first of all review the properties of the multivariate Gaussian distribution that make linear Gaussian models easier to deal with. We'll return to the, perhaps surprising, result on the parameters within the nonlinearity, \(\parameterVector\), shortly.

To work with linear Gaussian models, to find the marginal likelihood all you need to know is the following rules. If \[
\dataVector = \mappingMatrix \inputVector + \noiseVector,
\] where \(\dataVector\), \(\inputVector\) and \(\noiseVector\) are vectors and we assume that \(\inputVector\) and \(\noiseVector\) are drawn from multivariate Gaussians, \[\begin{align}
\inputVector & \sim \gaussianSamp{\meanVector}{\covarianceMatrix}\\
\noiseVector & \sim \gaussianSamp{\zerosVector}{\covarianceMatrixTwo}
\end{align}\] then we know that \(\dataVector\) is also drawn from a multivariate Gaussian with, \[
\dataVector \sim \gaussianSamp{\mappingMatrix\meanVector}{\mappingMatrix\covarianceMatrix\mappingMatrix^\top + \covarianceMatrixTwo}.
\] With apprioriately defined covariance, \(\covarianceTwoMatrix\), this is actually the marginal likelihood for Factor Analysis, or Probabilistic Principal Component Analysis (Tipping and Bishop, 1999), because we integrated out the inputs (or latent variables they would be called in that case).

However, we are focussing on what happens in models which are non-linear in the inputs, whereas the above would be linear in the inputs. To consider these, we introduce a matrix, called the design matrix. We set each activation function computed at each data point to be \[
\activationScalar_{i,j} = \activationScalar(\mappingVector^{(1)}_{j}, \inputVector_{i})
\] and define the matrix of activations (known as the design matrix in statistics) to be, \[
\activationMatrix =
\begin{bmatrix}
\activationScalar_{1, 1} & \activationScalar_{1, 2} & \dots & \activationScalar_{1, \numHidden} \\
\activationScalar_{1, 2} & \activationScalar_{1, 2} & \dots & \activationScalar_{1, \numData} \\
\vdots & \vdots & \ddots & \vdots \\
\activationScalar_{\numData, 1} & \activationScalar_{\numData, 2} & \dots & \activationScalar_{\numData, \numHidden}
\end{bmatrix}.
\] By convention this matrix always has \(\numData\) rows and \(\numHidden\) columns, now if we define the vector of all noise corruptions, \(\noiseVector = \left[\noiseScalar_1, \dots \noiseScalar_\numData\right]^\top\).

{ If we define the prior distribution over the vector \(\mappingVector\) to be Gaussian,} \[
\mappingVector \sim \gaussianSamp{\zerosVector}{\alpha\eye},
\]

{ then we can use rules of multivariate Gaussians to see that,} \[
\dataVector \sim \gaussianSamp{\zerosVector}{\alpha \activationMatrix \activationMatrix^\top + \dataStd^2 \eye}.
\]

In other words, our training data is distributed as a multivariate Gaussian, with zero mean and a covariance given by \[
\kernelMatrix = \alpha \activationMatrix \activationMatrix^\top + \dataStd^2 \eye.
\]

This is an \(\numData \times \numData\) size matrix. Its elements are in the form of a function. The maths shows that any element, index by \(i\) and \(j\), is a function only of inputs associated with data points \(i\) and \(j\), \(\dataVector_i\), \(\dataVector_j\). \(\kernel_{i,j} = \kernel\left(\inputVector_i, \inputVector_j\right)\)

If we look at the portion of this function associated only with \(\mappingFunction(\cdot)\), i.e. we remove the noise, then we can write down the covariance associated with our neural network, \[
\kernel_\mappingFunction\left(\inputVector_i, \inputVector_j\right) = \alpha \activationVector\left(\mappingMatrix_1, \inputVector_i\right)^\top \activationVector\left(\mappingMatrix_1, \inputVector_j\right)
\] so the elements of the covariance or kernel matrix are formed by inner products of the rows of the design matrix.

This is the essence of a Gaussian process. Instead of making assumptions about our density over each data point, \(\dataScalar_i\) as i.i.d. we make a joint Gaussian assumption over our data. The covariance matrix is now a function of both the parameters of the activation function, \(\mappingMatrixTwo\), and the input variables, \(\inputMatrix\). This comes about through integrating out the parameters of the model, \(\mappingVector\).

We can basically put anything inside the basis functions, and many people do. These can be deep kernels (Cho and Saul, 2009) or we can learn the parameters of a convolutional neural network inside there.

Viewing a neural network in this way is also what allows us to beform sensible batch normalizations (Ioffe and Szegedy, 2015).

Non-degenerate Gaussian Processes

The process described above is degenerate. The covariance function is of rank at most \(\numHidden\) and since the theoretical amount of data could always increase \(\numData \rightarrow \infty\), the covariance function is not full rank. This means as we increase the amount of data to infinity, there will come a point where we can't normalize the process because the multivariate Gaussian has the form, \[
\gaussianDist{\mappingFunctionVector}{\zerosVector}{\kernelMatrix} = \frac{1}{\left(2\pi\right)^{\frac{\numData}{2}}\det{\kernelMatrix}^\frac{1}{2}} \exp\left(-\frac{\mappingFunctionVector^\top\kernelMatrix \mappingFunctionVector}{2}\right)
\] and a non-degenerate kernel matrix leads to \(\det{\kernelMatrix} = 0\) defeating the normalization (it's equivalent to finding a projection in the high dimensional Gaussian where the variance of the the resulting univariate Gaussian is zero, i.e. there is a null space on the covariance, or alternatively you can imagine there are one or more directions where the Gaussian has become the delta function).

In the machine learning field, it was Radford Neal (Neal, 1994) that realized the potential of the next step. In his 1994 thesis, he was considering Bayesian neural networks, of the type we described above, and in considered what would happen if you took the number of hidden nodes, or neurons, to infinity, i.e. \(\numHidden \rightarrow \infty\).

Page 37 of Radford Neal's 1994 thesis

In loose terms, what Radford considers is what happens to the elements of the covariance function, \[
\begin{align*}
\kernel_\mappingFunction\left(\inputVector_i, \inputVector_j\right) & = \alpha \activationVector\left(\mappingMatrix_1, \inputVector_i\right)^\top \activationVector\left(\mappingMatrix_1, \inputVector_j\right)\\
& = \alpha \sum_k \activationScalar\left(\mappingVector^{(1)}_k, \inputVector_i\right) \activationScalar\left(\mappingVector^{(1)}_k, \inputVector_j\right)
\end{align*}
\] if instead of considering a finite number you sample infinitely many of these activation functions, sampling parameters from a prior density, \(p(\mappingVectorTwo)\), for each one, \[
\kernel_\mappingFunction\left(\inputVector_i, \inputVector_j\right) = \alpha \int \activationScalar\left(\mappingVector^{(1)}, \inputVector_i\right) \activationScalar\left(\mappingVector^{(1)}, \inputVector_j\right) p(\mappingVector^{(1)}) \text{d}\mappingVector^{(1)}
\] And that's not only for Gaussian \(p(\mappingVectorTwo)\). In fact this result holds for a range of activations, and a range of prior densities because of the central limit theorem.

To write it in the form of a probabilistic program, as long as the distribution for \(\phi_i\) implied by this short probabilistic program, \[
\begin{align*}
\mappingVectorTwo & \sim p(\cdot)\\
\phi_i & = \activationScalar\left(\mappingVectorTwo, \inputVector_i\right),
\end{align*}
\] has finite variance, then the result of taking the number of hidden units to infinity, with appropriate scaling, is also a Gaussian process.

Further Reading

To understand this argument in more detail, I highly recommend reading chapter 2 of Neal's thesis, which remains easy to read and clear today. Indeed, for readers interested in Bayesian neural networks, both Raford Neal's and David MacKay's PhD thesis (MacKay, 1992) remain essential reading. Both theses embody a clarity of thought, and an ability to weave together threads from different fields that was the business of machine learning in the 1990s. Radford and David were also pioneers in making their software widely available and publishing material on the web.

Exponentiated Quadratic Covariance

The exponentiated quadratic covariance, also known as the Gaussian covariance or the RBF covariance and the squared exponential. Covariance between two points is related to the negative exponential of the squared distnace between those points. This covariance function can be derived in a few different ways: as the infinite limit of a radial basis function neural network, as diffusion in the heat equation, as a Gaussian filter in Fourier space or as the composition as a series of linear filters applied to a base function.

The covariance takes the following form, [
\kernelScalar(\inputVector, \inputVector^\prime) = \alpha \exp\left(-\frac{\ltwoNorm{\inputVector - \inputVector^\prime}^2}{2\ell^2}\right)
] where (\ell) is the length scale or time scale of the process and (\alpha) represents the overall process variance.

Olympic Marathon Data

The first thing we will do is load a standard data set for regression modelling. The data consists of the pace of Olympic Gold Medal Marathon winners for the Olympics from 1896 to present. First we load in the data and plot.

Things to notice about the data include the outlier in 1904, in this year, the olympics was in St Louis, USA. Organizational problems and challenges with dust kicked up by the cars following the race meant that participants got lost, and only very few participants completed.

More recent years see more consistently quick marathons.

Our first objective will be to perform a Gaussian process fit to the data, we'll do this using the GPy software.

optimizes the parameters of the covariance function and the noise level of the model. Once the fit is complete, we'll try creating some test points, and computing the output of the GP model in terms of the mean and standard deviation of the posterior functions between 1870 and 2030. We plot the mean function and the standard deviation at 200 locations. We can obtain the predictions using

Fit Quality

In the fit we see that the error bars (coming mainly from the noise variance) are quite large. This is likely due to the outlier point in 1904, ignoring that point we can see that a tighter fit is obtained. To see this making a version of the model, m_clean, where that point is removed.

Data is fine for answering very specific questions, like "Who won the Olympic Marathon in 2012?", because we have that answer stored, however, we are not given the answer to many other questions. For example, Alan Turing was a formidable marathon runner, in 1946 he ran a time 2 hours 46 minutes (just under four minutes per kilometer, faster than I and most of the other Endcliffe Park Run runners can do 5 km). What is the probability he would have won an Olympics if one had been held in 1946?

Alan Turing, in 1946 he was only 11 minutes slower than the winner of the 1948 games. Would he have won a hypothetical games held in 1946? Source: Alan Turing Internet Scrapbook

Basis Function Covariance

The fixed basis function covariane just comes from the properties of a multivariate Gaussian, if we decide \[
\mappingFunctionVector=\basisMatrix\mappingVector
\] and then we assume \[
\mappingVector \sim \gaussianSamp{\zerosVector}{\alpha\eye}
\] then it follows from the properties of a multivariate Gaussian that \[
\mappingFunctionVector \sim \gaussianSamp{\zerosVector}{\alpha\basisMatrix\basisMatrix^\top}
\] meaning that the vector of observations from the function is jointly distributed as a Gaussian process and the covariance matrix is \(\kernelMatrix = \alpha\basisMatrix \basisMatrix^\top\), each element of the covariance matrix can then be found as the inner product between two rows of the basis funciton matrix. \[
\kernel(\inputVector, \inputVector^\prime) = \basisVector(\inputVector)^\top \basisVector(\inputVector^\prime)
\]

Brownian motion is also a Gaussian process. It follows a Gaussian random walk, with diffusion occuring at each time point driven by a Gaussian input. This implies it is both Markov and Gaussian. The covariane function for Brownian motion has the form [
\kernelScalar(t, t^\prime) = \alpha \min(t, t^\prime)
]

The covariance of Brownian motion, and some samples from the covariance showing the functional form.

The multi-layer perceptron covariance function. This is derived by considering the infinite limit of a neural network with probit activation functions.

\(=f\Bigg(\)\(\Bigg)\)

\(=f\Bigg(\)\(\Bigg)\)

The cosmic microwave background is, to a very high degree of precision, a Gaussian process. The parameters of its covariance function are given by fundamental parameters of the universe, such as the amount of dark matter and mass.

A Simple Regression Problem

Here we set up a simple one dimensional regression problem. The input locations, \(\inputMatrix\), are in two separate clusters. The response variable, \(\dataVector\), is sampled from a Gaussian process with an exponentiated quadratic covariance.

Now we set up the inducing variables, \(\mathbf{u}\). Each inducing variable has its own associated input index, \(\mathbf{Z}\), which lives in the same space as \(\inputMatrix\). Here we are using the true covariance function parameters to generate the fit.

A deep neural network. Input nodes are shown at the bottom. Each hidden layer is the result of applying an affine transformation to the previous layer and placing through an activation function.

Mathematically, each layer of a neural network is given through computing the activation function, \(\basisFunction(\cdot)\), contingent on the previous layer, or the inputs. In this way the activation functions, are composed to generate more complex interactions than would be possible with any single layer. \[
\begin{align}
\hiddenVector_{1} &= \basisFunction\left(\mappingMatrix_1 \inputVector\right)\\
\hiddenVector_{2} &= \basisFunction\left(\mappingMatrix_2\hiddenVector_{1}\right)\\
\hiddenVector_{3} &= \basisFunction\left(\mappingMatrix_3 \hiddenVector_{2}\right)\\
\dataVector &= \mappingVector_4 ^\top\hiddenVector_{3}
\end{align}
\]

Overfitting

One potential problem is that as the number of nodes in two adjacent layers increases, the number of parameters in the affine transformation between layers, \(\mappingMatrix\), increases. If there are \(k_{i-1}\) nodes in one layer, and \(k_i\) nodes in the following, then that matrix contains \(k_i k_{i-1}\) parameters, when we have layer widths in the 1000s that leads to millions of parameters.

One proposed solution is known as dropout where only a sub-set of the neural network is trained at each iteration. An alternative solution would be to reparameterize \(\mappingMatrix\) with its singular value decomposition. \[
\mappingMatrix = \eigenvectorMatrix\eigenvalueMatrix\eigenvectwoMatrix^\top
\] or \[
\mappingMatrix = \eigenvectorMatrix\eigenvectwoMatrix^\top
\] where if \(\mappingMatrix \in \Re^{k_1\times k_2}\) then \(\eigenvectorMatrix\in \Re^{k_1\times q}\) and \(\eigenvectwoMatrix \in \Re^{k_2\times q}\), i.e. we have a low rank matrix factorization for the weights.

import teaching_plots as plot

plot.low_rank_approximation(diagrams='../slides/diagrams')

Pictorial representation of the low rank form of the matrix \(\mappingMatrix\)

import teaching_plots as plot

plot.deep_nn_bottleneck(diagrams='../slides/diagrams/deepgp')

Including the low rank decomposition of \(\mappingMatrix\) in the neural network, we obtain a new mathematical form. Effectively, we are adding additional latent layers, \(\latentVector\), in between each of the existing hidden layers. In a neural network these are sometimes known as bottleneck layers. The network can now be written mathematically as \[
\begin{align}
\latentVector_{1} &= \eigenvectwoMatrix^\top_1 \inputVector\\
\hiddenVector_{1} &= \basisFunction\left(\eigenvectorMatrix_1 \latentVector_{1}\right)\\
\latentVector_{2} &= \eigenvectwoMatrix^\top_2 \hiddenVector_{1}\\
\hiddenVector_{2} &= \basisFunction\left(\eigenvectorMatrix_2 \latentVector_{2}\right)\\
\latentVector_{3} &= \eigenvectwoMatrix^\top_3 \hiddenVector_{2}\\
\hiddenVector_{3} &= \basisFunction\left(\eigenvectorMatrix_3 \latentVector_{3}\right)\\
\dataVector &= \mappingVector_4^\top\hiddenVector_{3}.
\end{align}
\]

Equivalent to prior over parameters, take width of each layer to infinity.

Mathematically, a deep Gaussian process can be seen as a composite multivariate function, \[
\mathbf{g}(\inputVector)=\mappingFunctionVector_5(\mappingFunctionVector_4(\mappingFunctionVector_3(\mappingFunctionVector_2(\mappingFunctionVector_1(\inputVector))))).
\] Or if we view it from the probabilistic perspective we can see that a deep Gaussian process is specifying a factorization of the joint density, the standard deep model takes the form of a Markov chain.

Why Deep?

If the result of composing many functions together is simply another function, then why do we bother? The key point is that we can change the class of functions we are modeling by composing in this manner. A Gaussian process is specifying a prior over functions, and one with a number of elegant properties. For example, the derivative process (if it exists) of a Gaussian process is also Gaussian distributed. That makes it easy to assimilate, for example, derivative observations. But that also might raise some alarm bells. That implies that the marginal derivative distribution is also Gaussian distributed. If that's the case, then it means that functions which occasionally exhibit very large derivatives are hard to model with a Gaussian process. For example, a function with jumps in.

A one off discontinuity is easy to model with a Gaussian process, or even multiple discontinuities. They can be introduced in the mean function, or independence can be forced between two covariance functions that apply in different areas of the input space. But in these cases we will need to specify the number of discontinuities and where they occur. In otherwords we need to parameterise the discontinuities. If we do not know the number of discontinuities and don't wish to specify where they occur, i.e. if we want a non-parametric representation of discontinuities, then the standard Gaussian process doesn't help.

Stochastic Process Composition

The deep Gaussian process leads to non-Gaussian models, and non-Gaussian characteristics in the covariance function. In effect, what we are proposing is that we change the properties of the functions we are considering by *composing stochastic processes$. This is an approach to creating new stochastic processes from well known processes.

Additionally, we are not constrained to the formalism of the chain. For example, we can easily add single nodes emerging from some point in the depth of the chain. This allows us to combine the benefits of the graphical modelling formalism, but with a powerful framework for relating one set of variables to another, that of Gaussian processes

Olympic Marathon Data

The first thing we will do is load a standard data set for regression modelling. The data consists of the pace of Olympic Gold Medal Marathon winners for the Olympics from 1896 to present. First we load in the data and plot.

Things to notice about the data include the outlier in 1904, in this year, the olympics was in St Louis, USA. Organizational problems and challenges with dust kicked up by the cars following the race meant that participants got lost, and only very few participants completed.

More recent years see more consistently quick marathons.

Our first objective will be to perform a Gaussian process fit to the data, we'll do this using the GPy software.

optimizes the parameters of the covariance function and the noise level of the model. Once the fit is complete, we'll try creating some test points, and computing the output of the GP model in terms of the mean and standard deviation of the posterior functions between 1870 and 2030. We plot the mean function and the standard deviation at 200 locations. We can obtain the predictions using

Fit Quality

In the fit we see that the error bars (coming mainly from the noise variance) are quite large. This is likely due to the outlier point in 1904, ignoring that point we can see that a tighter fit is obtained. To see this making a version of the model, m_clean, where that point is removed.

Data is fine for answering very specific questions, like "Who won the Olympic Marathon in 2012?", because we have that answer stored, however, we are not given the answer to many other questions. For example, Alan Turing was a formidable marathon runner, in 1946 he ran a time 2 hours 46 minutes (just under four minutes per kilometer, faster than I and most of the other Endcliffe Park Run runners can do 5 km). What is the probability he would have won an Olympics if one had been held in 1946?

Alan Turing, in 1946 he was only 11 minutes slower than the winner of the 1948 games. Would he have won a hypothetical games held in 1946? Source: Alan Turing Internet Scrapbook

Deep GP Fit

Let's see if a deep Gaussian process can help here. We will construct a deep Gaussian process with one hidden layer (i.e. one Gaussian process feeding into another).

Build a Deep GP with an additional hidden layer (one dimensional) to fit the model.

Deep Gaussian process models also can require some thought in initialization. Here we choose to start by setting the noise variance to be one percent of the data variance.

Optimization requires moving variational parameters in the hidden layer representing the mean and variance of the expected values in that layer. Since all those values can be scaled up, and this only results in a downscaling in the output of the first GP, and a downscaling of the input length scale to the second GP. It makes sense to first of all fix the scales of the covariance function in each of the GPs.

Sometimes, deep Gaussian processes can find a local minima which involves increasing the noise level of one or more of the GPs. This often occurs because it allows a minimum in the KL divergence term in the lower bound on the likelihood. To avoid this minimum we habitually train with the likelihood variance (the noise on the output of the GP) fixed to some lower value for some iterations.

Let's create a helper function to initialize the models we use in the notebook.

Finally, we allow the noise variance to change and optimize for a large number of iterations.

for layer in m.layers:
layer.likelihood.variance.constrain_positive(warning=False)
m.optimize(messages=True,max_iters=10000)

For our optimization process we define a new function.

def staged_optimize(self, iters=(1000,1000,10000), messages=(False, False, True)):
"""Optimize with parameters constrained and then with parameters released"""for layer inself.layers:
# Fix the scale of each of the covariance functions.
layer.kern.variance.fix(warning=False)
layer.kern.lengthscale.fix(warning=False)
# Fix the variance of the noise in each layer.
layer.likelihood.variance.fix(warning=False)
self.optimize(messages=messages[0],max_iters=iters[0])
for layer inself.layers:
layer.kern.lengthscale.constrain_positive(warning=False)
self.obslayer.kern.variance.constrain_positive(warning=False)
self.optimize(messages=messages[1],max_iters=iters[1])
for layer inself.layers:
layer.kern.variance.constrain_positive(warning=False)
layer.likelihood.variance.constrain_positive(warning=False)
self.optimize(messages=messages[2],max_iters=iters[2])
# Bind the new method to the Deep GP object.
deepgp.DeepGP.staged_optimize=staged_optimize

m.staged_optimize(messages=(True,True,True))

Plot the prediction

The prediction of the deep GP can be extracted in a similar way to the normal GP. Although, in this case, it is an approximation to the true distribution, because the true distribution is not Gaussian.

Olympic Marathon Pinball Plot

The pinball plot shows the flow of any input ball through the deep Gaussian process. In a pinball plot a series of vertical parallel lines would indicate a purely linear function. For the olypmic marathon data we can see the first layer begins to shift from input towards the right. Note it also does so with some uncertainty (indicated by the shaded backgrounds). The second layer has less uncertainty, but bunches the inputs more strongly to the right. This input layer of uncertainty, followed by a layer that pushes inputs to the right is what gives the heteroschedastic noise.

Fit a Deep GP

We're going to fit a Deep Gaussian process model to the MNIST data with two hidden layers. Each of the two Gaussian processes (one from the first hidden layer to the second, one from the second hidden layer to the data) has an exponentiated quadratic covariance.

Initialization

Just like deep neural networks, there are some tricks to intitializing these models. The tricks we use here include some early training of the model with model parameters constrained. This gives the variational inducing parameters some scope to tighten the bound for the case where the noise variance is small and the variances of the Gaussian processes are around 1.

Uncertainty Quantification

Proposal: Deep GPs may also be a great approach, but better to deploy according to natural strengths.

Uncertainty Quantification

Probabilistic numerics, surrogate modelling, emulation, and UQ.

Not a fan of AI as a term.

But we are faced with increasing amounts of algorithmic decision making.

ML and Decision Making

When trading off decisions: compute or acquire data?

There is a critical need for uncertainty.

Uncertainty Quantification

Uncertainty quantification (UQ) is the science of quantitative characterization and reduction of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known.

Interaction between physical and virtual worlds of major interest for Amazon.

We will to illustrate different concepts of Uncertainty Quantification (UQ) and the role that Gaussian processes play in this field. Based on a simple simulator of a car moving between a valley and a mountain, we are going to illustrate the following concepts:

Systems emulation. Many real world decisions are based on simulations that can be computationally very demanding. We will show how simulators can be replaced by emulators: Gaussian process models fitted on a few simulations that can be used to replace the simulator. Emulators are cheap to compute, fast to run, and always provide ways to quantify the uncertainty of how precise they are compared the original simulator.

Emulators in optimization problems. We will show how emulators can be used to optimize black-box functions that are expensive to evaluate. This field is also called Bayesian Optimization and has gained an increasing relevance in machine learning as emulators can be used to optimize computer simulations (and machine learning algorithms) quite efficiently.

Multi-fidelity emulation methods. In many scenarios we have simulators of different quality about the same measure of interest. In these cases the goal is to merge all sources of information under the same model so the final emulator is cheaper and more accurate than an emulator fitted only using data from the most accurate and expensive simulator.

Example: Formula One Racing

Designing an F1 Car requires CFD, Wind Tunnel, Track Testing etc.

How to combine them?

Mountain Car Simulator

To illustrate the above mentioned concepts we we use the mountain car simulator. This simulator is widely used in machine learning to test reinforcement learning algorithms. The goal is to define a control policy on a car whose objective is to climb a mountain. Graphically, the problem looks as follows:

The goal is to define a sequence of actions (push the car right or left with certain intensity) to make the car reach the flag after a number \(T\) of time steps.

At each time step \(t\), the car is characterized by a vector \(\inputVector_{t} = (p_t,v_t)\) of states which are respectively the the position and velocity of the car at time \(t\). For a sequence of states (an episode), the dynamics of the car is given by

where \(\textbf{u}_{t}\) is the value of an action force, which in this example corresponds to push car to the left (negative value) or to the right (positive value). The actions across a full episode are represented in a policy \(\textbf{u}_{t} = \pi(\inputVector_{t},\theta)\) that acts according to the current state of the car and some parameters \(\theta\). In the following examples we will assume that the policy is linear which allows us to write \(\pi(\inputVector_{t},\theta)\) as

\[\pi(\inputVector,\theta)= \theta_0 + \theta_p p + \theta_vv.\]

For \(t=1,\dots,T\) now given some initial state \(\inputVector_{0}\) and some some values of each \(\textbf{u}_{t}\), we can simulate the full dynamics of the car for a full episode using Gym. The values of \(\textbf{u}_{t}\) are fully determined by the parameters of the linear controller.

After each episode of length \(T\) is complete, a reward function \(R_{T}(\theta)\) is computed. In the mountain car example the reward is computed as 100 for reaching the target of the hill on the right hand side, minus the squared sum of actions (a real negative to push to the left and a real positive to push to the right) from start to goal. Note that our reward depend on \(\theta\) as we make it dependent on the parameters of the linear controller.

Emulate the Mountain Car

import gym

env = gym.make('MountainCarContinuous-v0')

Our goal in this section is to find the parameters \(\theta\) of the linear controller such that

\[\theta^* = arg \max_{\theta} R_T(\theta).\]

In this section, we directly use Bayesian optimization to solve this problem. We will use GPyOpt so we first define the objective function:

For each set of parameter values of the linear controller we can run an episode of the simulator (that we fix to have a horizon of \(T=500\)) to generate the reward. Using as input the parameters of the controller and as outputs the rewards we can build a Gaussian process emulator of the reward.

In Bayesian optimization an acquisition function is used to balance exploration and exploitation to evaluate new locations close to the optimum of the objective. In this notebook we select the expected improvement (EI). For further details have a look to the review paper of Shahriari et al (2015).

As we can see the random linear controller does not manage to push the car to the top of the mountain. Now, let's optimize the regret using Bayesian optimization and the emulator for the reward. We try 50 new parameters chosen by the EI.

he car can now make it to the top of the mountain! Emulating the reward function and using the EI helped as to find a linear controller that solves the problem.

Data Efficient Emulation

In the previous section we solved the mountain car problem by directly emulating the reward but no considerations about the dynamics \(\inputVector_{t+1} = \mappingFunction(\inputVector_{t},\textbf{u}_{t})\) of the system were made. Note that we had to run 75 episodes of 500 steps each to solve the problem, which required to call the simulator \(500\times 75 =37500\) times. In this section we will show how it is possible to reduce this number by building an emulator for \(f\) that can later be used to directly optimize the control.

The inputs of the model for the dynamics are the velocity, the position and the value of the control so create this space accordingly.

Next, we sample some input parameters and use the simulator to compute the outputs. Note that in this case we are not running the full episodes, we are just using the simulator to compute \(\inputVector_{t+1}\) given \(\inputVector_{t}\) and \(\textbf{u}_{t}\).

In general we might use much smarter strategies to design our emulation of the simulator. For example, we could use the variance of the predictive distributions of the models to collect points using uncertainty sampling, which will give us a better coverage of the space. For simplicity, we move ahead with the 500 randomly selected points.

Now that we have a data set, we can update the emulators for the location and the velocity.

We can now have a look to how the emulator and the simulator match. First, we show a contour plot of the car aceleration for each pair of can position and velocity. You can use the bar bellow to play with the values of the controler to compare the emulator and the simulator.

We can see how the emulator is doing a fairly good job approximating the simulator. On the edges, however, it struggles to captures the dynamics of the simulator.

Given some input parameters of the linear controlling, how do the dynamics of the emulator and simulator match? In the following figure we show the position and velocity of the car for the 500 time steps of an episode in which the parameters of the linear controller have been fixed beforehand. The value of the input control is also shown.

We now make explicit use of the emulator, using it to replace the simulator and optimize the linear controller. Note that in this optimization, we don't need to query the simulator anymore as we can reproduce the full dynamics of an episode using the emulator. For illustrative purposes, in this example we fix the initial location of the car.

And the problem is again solved, but in this case we have replaced the simulator of the car dynamics by a Gaussian process emulator that we learned by calling the simulator only 500 times. Compared to the 37500 calls that we needed when applying Bayesian optimization directly on the simulator this is a great gain.

In some scenarios we have simulators of the same environment that have different fidelities, that is that reflect with different level of accuracy the dynamics of the real world. Running simulations of the different fidelities also have a different cost: hight fidelity simulations are more expensive the cheaper ones. If we have access to these simulators we can combine high and low fidelity simulations under the same model.

So let's assume that we have two simulators of the mountain car dynamics, one of high fidelity (the one we have used) and another one of low fidelity. The traditional approach to this form of multi-fidelity emulation is to assume that

where \(\mappingFunction_{i-1}\left(\inputVector\right)\) is a low fidelity simulation of the problem of interest and \(\mappingFunction_i\left(\inputVector\right)\) is a higher fidelity simulation. The function \(\delta_i\left(\inputVector \right)\) represents the difference between the lower and higher fidelity simulation, which is considered additive. The additive form of this covariance means that if \(\mappingFunction_{0}\left(\inputVector\right)\) and \(\left\{\delta_i\left(\inputVector \right)\right\}_{i=1}^m\) are all Gaussian processes, then the process over all fidelities of simuation will be a joint Gaussian process.

where the low fidelity representation is non linearly transformed by \(\mappingFunctionTwo(\cdot)\) before use in the process. This is the approach taken in Perdikaris et al. (2017). But once we accept that these models can be composed, a highly flexible framework can emerge. A key point is that the data enters the model at different levels, and represents different aspects. For example these correspond to the two fidelities of the mountain car simulator.

In classical statistics we often interpret these parameters, \(\beta\), whereas in machine learning we are normally more interested in the result of the prediction, and less in the prediction. Although this is changing with more need for accountability. In honour of this I normally use \(\boldsymbol{\beta}\) when I care about the value of these parameters, and \(\mappingVector\) when I care more about the quality of the prediction.↩